Problem: A few families took a trip to an amusement park together. Tickets cost $$7.50$ each for adults and $$3.50$ each for kids, and the group paid $$39.50$ in total. There were $5$ fewer adults than kids in the group. Find the number of adults and kids on the trip.
Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${7.5x+3.5y = 39.5}$ ${x = y-5}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-5}$ for $x$ in the first equation. ${7.5}{(y-5)}{+ 3.5y = 39.5}$ Simplify and solve for $y$ $ 7.5y-37.5 + 3.5y = 39.5 $ $ 11y-37.5 = 39.5 $ $ 11y = 77 $ $ y = \dfrac{77}{11} $ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into ${x = y-5}$ to find $x$ ${x = }{(7)}{ - 5}$ ${x = 2}$ You can also plug ${y = 7}$ into ${7.5x+3.5y = 39.5}$ and get the same answer for $x$ ${7.5x + 3.5}{(7)}{= 39.5}$ ${x = 2}$ There were $2$ adults and $7$ kids.